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Mirrors > Home > MPE Home > Th. List > cnmsgnbas | Structured version Visualization version GIF version |
Description: The base set of the sign subgroup of the complex numbers. (Contributed by Stefan O'Rear, 28-Aug-2015.) |
Ref | Expression |
---|---|
cnmsgngrp.u | ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) |
Ref | Expression |
---|---|
cnmsgnbas | ⊢ {1, -1} = (Base‘𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 10685 | . . 3 ⊢ 1 ∈ ℂ | |
2 | neg1cn 11842 | . . 3 ⊢ -1 ∈ ℂ | |
3 | prssi 4719 | . . 3 ⊢ ((1 ∈ ℂ ∧ -1 ∈ ℂ) → {1, -1} ⊆ ℂ) | |
4 | 1, 2, 3 | mp2an 692 | . 2 ⊢ {1, -1} ⊆ ℂ |
5 | cnmsgngrp.u | . . 3 ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
6 | eqid 2739 | . . . 4 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
7 | cnfldbas 20233 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
8 | 6, 7 | mgpbas 19376 | . . 3 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
9 | 5, 8 | ressbas2 16670 | . 2 ⊢ ({1, -1} ⊆ ℂ → {1, -1} = (Base‘𝑈)) |
10 | 4, 9 | ax-mp 5 | 1 ⊢ {1, -1} = (Base‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2114 ⊆ wss 3853 {cpr 4528 ‘cfv 6349 (class class class)co 7182 ℂcc 10625 1c1 10628 -cneg 10961 Basecbs 16598 ↾s cress 16599 mulGrpcmgp 19370 ℂfldccnfld 20229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-7 11796 df-8 11797 df-9 11798 df-n0 11989 df-z 12075 df-dec 12192 df-uz 12337 df-fz 12994 df-struct 16600 df-ndx 16601 df-slot 16602 df-base 16604 df-sets 16605 df-ress 16606 df-plusg 16693 df-mulr 16694 df-starv 16695 df-tset 16699 df-ple 16700 df-ds 16702 df-unif 16703 df-mgp 19371 df-cnfld 20230 |
This theorem is referenced by: psgnghm 20408 psgninv 20410 psgnodpm 20416 |
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